A Parallel Architecture for the Generalized Traveling Salesman Problem

1
A Parallel Architecture for the Generalized
Traveling Salesman Problem

• Max Scharrenbroich
• AMSC 663 Mid-Year Progress Report
• Advisor Dr. Bruce L. Golden
• R. H. Smith School of Business

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Presentation Overview

• Background and Review
• GTSP Review
• Applications
• Problem Formulation and Algorithms
• Review of Project Objectives
• Review of mrOX GA
• Parallelism in the mrOX GA
• Status Summary and Future Work

Review gt
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Review of the GTSP

• The Generalized Traveling Salesman Problem
  (GTSP)
• Variation of the well-known traveling salesman
  problem.
• A set of nodes is partitioned into a number of
  clusters.
• Objective Find a minimum-cost tour visiting
  exactly one node in each cluster.
• Example on the following slides

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GTSP Example

• Start with a set of nodes or locations to visit.

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GTSP Example (continued)

• Partition the nodes into clusters.

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GTSP Example (continued)

• Find the minimum tour visiting each cluster.

Applications gt
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Applications

• The GTSP has many real-world applications in the
  field of routing
• Mailbox collection and stochastic vehicle
  routing.
• Warehouse order picking with multiple stock
  locations.
• Airport selection and routing for courier planes.

Formulation and Algorithms gt
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Problem Formulation and Algorithms

• The GTSP can be formulated as a 0-1 Integer
  Linear Program (ILP).
• The GTSP is an NP-hard problem.
• There are algorithms for solving the GTSP to
  optimality, but these algorithms eventually
  suffer from combinatorial explosion (gt90
  clusters).
• There are several successful heuristic approaches
  to the GTSP.
• In this project I will focus on the mrOX Genetic
  Algorithm (J. Silberholz and B.L. Golden, 2007).

Review of Project Objectivesgt
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Review of Project Objectives

• Develop a generic software architecture and
  framework for parallelizing serial heuristics for
  combinatorial optimization.
• Extend the framework to host the serial mrOX GA
  and the GTSP problem class.
• Investigate the performance of the parallel
  implementation of the mrOX GA on large instances
  of the GTSP (gt 90 clusters).

Overview gt
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Presentation Overview

• Background and Review
• Review of mrOX GA
• mrOX Crossover
• Local Search 2-opt and 1-swap
• Mutation
• Parallelism in the mrOX GA
• Status Summary and Future Work

Diagram of mrOX GA gt
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Review of mrOX GA
Start
Pop 1
Pop 2

Pop 7
Merge
Post-Merge
End
Crossover and Local Search gt
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Review of mrOX GA
mrOX Crossover gt
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Review of mrOX Crossover
Example gt
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Example of mrOX Crossover
Complexity gt
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Complexity of mrOX Crossover
Local Search gt
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Local Search 2-opt 1-swap
Mutation gt
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Mutation
(1) Randomly generate cut points.
Overview gt
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Presentation Overview

• Background and Review
• Review of mrOX GA
• Parallelism in the mrOX GA
• Low Level Parallelism
• Concurrent Exploration
• Cellular GA Inspired Parallel Cooperation
• Status Summary and Future Work

Low Level Parallelism gt
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Low-Level Parallelism in the mrOX GA

• For rOX and mrOX computational loading can be
  estimated (slide 13) and therefore crossover of
  individuals could be load balanced over a number
  of processors.
• The local search improvement phase (multiple
  cycles of 2-opt followed by 1-swap) in the
  post-merge phase is not deterministic and would
  be difficult to load balance.
• While improving execution time, load balancing
  the crossover and local search would introduce
  inefficiencies.
• Tuning the load balancing would be problem
  dependent and time consuming .

Type 3 Parallelism gt
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Type 3 Parallelism in the mrOX GA

• Genetic algorithms are amenable to parallelism
  via concurrent exploration.
• Cooperation between processes can be implemented
  to ensure diversity while maintaining
  intensification.

Parallel Cooperation w/ Mesh gt
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Parallel Cooperation with Mesh Topology

• Inspired by cellular genetic algorithms (cGAs),
  where individuals in a population only interact
  with nearest neighbors.
• Processes cooperate over a toroidal mesh
  topology.
• Ensures diversity while maintaining
  intensification.

Each process has four neighbors.
Processes periodically exchange the best
solutions with neighbors.
High-quality solutions diffuse through the
population.
Overview gt
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Presentation Overview

• Background and Review
• Review of mrOX GA
• Parallelism in the mrOX GA
• Status Summary and Future Work

Status Summary gt
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Status Summary

• Investigated different ways of using parallelism
  in the mrOX GA.
• Learned about cellular genetic algorithms (cGAs)
  as a motivation for parallel cooperation schemes.
• Completed a preliminary software design and began
  coding.
• Coded and ran an intermediate test application to
  validate mesh communication pattern.

Future Work gt
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Future Work

• Obtain a user account on the Deepthought cluster.
• Continue with design and coding.
• Performance testing of parallel implementation.

End gt
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References

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  generalized traveling salesman problem.
  Operations Research 45 (3) 378394, 1997.
• G. Laporte, A. Asef-Vaziri, C. Sriskandarajah.
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  Salesman Problem. Journal of the Operational
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• C.E. Noon. The generalized traveling salesman
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  Michigan, 1988.
• C.E. Noon. A Lagrangian based approach for the
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• J.P. Saksena. Mathematical model of scheduling
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  185-200, 1970.
• J. Silberholz and B.L. Golden. The Generalized
  Traveling Salesman Problem A New Genetic
  Algorithm Approach. Operations Research/Computer
  Science Interfaces Series 37 165-181, 2007.
• L. Snyder and M. Daskin. A random-key genetic
  algorithm for the generalized traveling salesman
  problem. European Journal of Operational Research
  17 (1) 38-53, 2006.

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Acknowledgements

• Chris Groer, University of Maryland
• William Mennell, University of Maryland
• John Silberholz, University of Maryland